Large anisotropy of electrical conductivity induced high thermoelectric performance of p-type CrSi2

Journal of Alloys and Compounds 581 (2013) 413–417

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Large anisotropy of electrical conductivity induced high thermoelectric performance of p-type CrSi2 Xiao Jing Zhang a, Yu Li Yan a, Yuan Xu Wang a,b,⇑ a b

Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Institute of Applied Physics, Guizhou Normal College, Guiyang 550018, Peoples Republic of China

a r t i c l e

i n f o

Article history: Received 2 February 2013 Received in revised form 1 July 2013 Accepted 2 July 2013 Available online 13 July 2013 Keywords: Transition-metal silicides Electronic structure Thermoelectric properties

a b s t r a c t A comparative study of thermoelectric performances about CrSi2 and b-FeSi2 were performed by using density functional theory and Boltzmann transport theory. It is found that the transport properties of p-type CrSi2 could be better than that of n-type. The high thermoelectric performance of p-type CrSi2 are possibly due to the high anisotropy of its electrical conductivity with p-type doping. For CrSi2, the effective mass of holes along the z direction is smaller than that along the x direction, and consequently the hole mobility along the z direction may be higher than that along the x direction. A high thermoelectric performance of CrSi2 could be achieved by hole doping with concentration range of 1–6  1021 cm3. The transport properties of n-type b-FeSi2 may be better than p-type one. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Thermoelectric materials can convert heat into electricity without using any moving parts or environmentally harmful fluids. They can easily extract energy from waste heat streams or other low-grade sources of energy. Great efforts have been devoted to searching high performance thermoelectrics. The key issue for achieving high thermoelectric performance is to increase thermoelectric figure of merit zT = S2rT/(je + jl), where S is the Seebeck coefficient (also known as the thermopower), r is the electrical conductivity, je and jl are the electronic and lattice thermal conductivity, and T is the absolute temperature [1]. Therefore, good thermoelectric materials should have a large Seebeck coefficient, a high electrical conductivity, and a low thermal conductivity. Metals always have very high electrical conductivity, but high thermal conductivity. Glasses have low thermal conductivity, but very low electrical conductivity. Therefore, ideal thermoelectric material should exhibit ‘‘phonon glass and electron crystal’’ behavior [2–4]. Transition metal (TM) silicides have various potential high-temperature applications due to their high melting points, rich resources, low price, and chemical stability at elevated temperatures [5–8]. CrSi2 is a narrow-band-gap (0.3–0.35 eV) semiconductor and its ZT value is only 0.2 at 450 K [9,10]. The low ZT of CrSi2 is thought to be caused by its small band gap. Small band gap always leads to bipolar effect on electrical conductivity, ⇑ Corresponding author at: Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China. Tel./fax: +86 378 3881488. E-mail address: [email protected] (Y.X. Wang). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.07.020

which is detrimental to high thermopower. However, Parker and Singh [11] reported that high-temperature thermoelectric performance of CrSi2 can be improved by electron doping (1– 4  1021 cm3). They found that heavy conduction band mass may lead to the high thermopower of n-type doping in CrSi2, which does not appear in its valence bands. Here, our electronic structure and Boltzmann transport calculations show that for CrSi2, p-type doping may bring to higher thermoelectric performance than ntype doping. High anisotropy of electrical conductivity of CrSi2 may result in a high thermoelectric performance along z direction of p-type doping. Extensive experimental works and theoretical calculations have been performed to study optical properties, electronic structure, and thermoelectric performances of b-FeSi2 [12– 17]. In this work, we also investigate the thermoelectric properties of b-FeSi2 for comparison.

2. Computational detail The structures of CrSi2 and b-FeSi2 were optimized by the Vienna ab initio simulation package (VASP) [18–20]. We used the projector augmented wave (PAW) method of Blöchl [21] in the implementation of Kresse and Joubert [22]. The Perdew–Burke– Ernzerhof generalized-gradient approximation (PBE-GGA) [23] was used for the exchange correlation potential. Plane-wave cutoff energy of 500 eV. For the Brillouin zone integrations of CrSi2 and bFeSi2, the k-point sampling 13  13  8 points and 5  6  6 points were chosen, respectively. Considering the positions of atoms and lattice constants, the Hellmann–Feynman forces on each ion are less than 0.005 eV/Å.

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The electronic structures of CrSi2 and b-FeSi2 were then calculated by the full-potential linearized augmented plane waves method [24], as implemented in the WIEN2k [25–27]. The Engel– Vosko [28] with generalized-gradient approximation (EV-GGA) was used for the exchange correlation potential, which can provide an accurate band gap. An accurate band gap is very important for evaluating thermoelectric performance. The muffin-tin radii were chosen to be 2.46, 2.18, 2.32 for Cr, Si, Fe, respectively. Cutoff parameter Rmt  Kmax = 7 (Kmax is the magnitude of the largest k vector) and the self-consistent calculations k points in Brillouin are 1000. Seebeck coefficient and electrical conductivity were calculated by using the semiclassical Boltzmann theory and rigidband approach [29]. 3. Results and discussions Fig. 1 shows the optimized structures. The left pattern is CrSi2 crystal which has a C40 hexagonal structure with space group P6222, and there are three Cr atoms and six Si atoms in each unit cell. The right pattern is b-FeSi2 crystal which has an orthorhombic structure with space group Cmca, and there are sixteen Fe atoms and thirty-two Si atoms in each unit cell. For CrSi2, the calculated lattice constants are a = 4.4069 Å and c = 6.3647 Å. For b-FeSi2, the calculated lattice constants are a = 9.8802 Å, b = 7.7692 Å, and c = 7.8140 Å. Those calculated lattice parameters of CrSi2 and bFeSi2 are close to the experimental values [30,31]. The transport properties of materials can by analyzed from their band structure. An anisotropic electronic structure has an important effect on thermoelectric properties. For the dependencies of electronic structure and the figure of merit, Hicks and Dresselahaus [32] suggested that in an anisotropic three-dimensional singleband case, the figure of merit ZT increases with an inherent parameter B when the electrical and thermal currents travel in the same direction, the inherent parameter B defined as follow:

B¼

 3=2 2 1 2kB T    1=2 kB T l ðm m m Þ ; x y z 2 3p 2 ejl h

ð1Þ

where mi ði ¼ ðx; y; zÞÞ is the effective mass of the carriers (electrons or holes) in the ith direction, l is the carrier mobility along the transport direction, and jl is the lattice thermal conductivity. The maximum attainable figure of merit (Zmax) of the formula:

Z max / c

T 3=2 sz

qffiffiffiffiffiffiffiffiffiffi ffi  

jl

mx my mz

eðrþ0:5Þ ;

ð2Þ

where c is the degeneracy of band extrema, sz is the relaxation time of the carriers moving along the transport (z) direction, r is the scattering parameter.

Fig. 1. Crystal structure of CrSi2 (left), black and blue spheres represent Cr and Si atoms, respectively. Crystal structure of b-FeSi2 (right), black and blue spheres represent Fe and Si atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Eqs. (1) and (2) suggest that high ZT value is benefited from a large effective mass, a high carrier mobility, a low lattice thermal conductivity, and high degeneracy of band extrema. Band structure calculation can give us informations of these properties directly. So it is necessary to analyze electronic structures of CrSi2 and b-FeSi2. Fig. 2 shows the calculated band structures of CrSi2 and b-FeSi2, and their first Brillouin zone is shown in Fig. 3. Because transport properties are closely related to the electronic states near the highest valence band and the lowest conduction band, we only focus on the electronic states near the Fermi level. The left panel of Fig. 2 is the calculated band structure of CrSi2. As seen in this figure, CrSi2 is a semiconductor with an indirect gap of 0.4 eV. The valence bands contains one main maximum located at the L point. Moreover, light and heavy bands appears together around the Fermi level, which indicates that CrSi2 may have a good thermoelectric performance. The right panel of Fig. 2 is the calculated band structure of b-FeSi2. b-FeSi2 is a semiconductor with an indirect gap of 0.79 eV. The valence band contains one main maximum located at the Y point, and the degeneracy of conduction band extrema is large. Thus, b-FeSi2 may also have a good thermoelectric performance. The transport properties of CrSi2 and b-FeSi2 were calculated based on the calculated EV-GGA electronic structures by using 2 the Boltzmann theory. Fig. 4 shows the calculated S; n; rs, and S sr r as a function of temperature, s is the electrical conductivity relative 2 to relaxation time, S sr is the powerfactor with respect to relaxation time. Fig. 4(a) shows that the sign of S for CrSi2 over the entire studied temperature range is positive, and the largest value of S is 240 lV/K. S drops above 600 K, and the possible reason is the increasing of carrier concentration above 600 K, S is inversely proportional to n. The temperature dependence of the carrier concentration for CrSi2 is shown in Fig. 4(b), and we can clearly see an increasing carrier concentration with increasing temperature. Fig. 4(c) shows that the temperature dependence of rs for CrSi2. 2 As seen from Fig. 4(d), the magnitude of S sr rapidly increases under 1000 K with increasing temperature, and then does gradually. The 2 maximum of S sr for CrSi2 is 6.7  1011 W/K2 ms at 1400 K. As seen in Fig. 4(e), the sign of S for b-FeSi2 over entire temperature range is also positive, and the largest value of S is 220 lV/K, which is larger than that of good thermoelectric material Bi2Te3 (200 lV/K at 300 K) [33]. The high S indicate that CrSi2 and b-FeSi2 are potential good thermoelectric materials. The temperature dependence of the carrier concentration for b-FeSi2 is shown in Fig. 4(f). From Fig. 4(b) and (f), we can clearly see that the carrier concentration increases gradually with increasing temperature at low temperature, then increases rapidly at the high temperature. In addition, the n values of CrSi2 are larger than those of b-FeSi2. Above phenomena can be explained as follow: temperature and band gap can strongly affect carrier concentration. As temperature increases, thermal excitation becomes strong, and consequently intrinsic carrier concentration increases. Since the smaller band gap of CrSi2 than b-FeSi2, its electrons could be easily excited from the valence band to the conduction band at same temperature, so the higher intrinsic carrier concentration may appears in CrSi2. Fig. 4(g) shows that the temperature dependence of rs for b-FeSi2. As seen in Fig. 4(c) and (g), r increases with increasing temperature, revealing of thermal actis vation of carriers across the band gap, typical of semiconductors. Below 400 K, rs of CrSi2 and b-FeSi2 are similar. Above 400 K, rs of CrSi2 is always larger than that of b-FeSi2, which mainly comes from the higher concentration of CrSi2 than b-FeSi2. From 2 Fig. 4(h), the temperature dependence of S sr of b-FeSi2 is positive. S2 r The magnitude of s gradually increases with increasing temperature, and the maximum value about 4  1011 W/K2 ms may be attainable at 1500 K. Next, we turn to study p-type and n-type doping effect on thermoelectric properties of CrSi2 and b-FeSi2 at 300 K, 800 K, 1000 K, 1500 K, respectively. We found that the transport properties along

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Γ

Μ

Η Κ

Α

Γ

Γ

Α

Ζ

Ζ

Γ

Fig. 2. Calculated band structures of CrSi2 (left) b-FeSi2 (right).

Fig. 3. The first Brillouin zone (from the primitive unit cell) of CrSi2 (left) b-FeSi2 (right).

the y direction are close to those along the x direction. Fig. 5 shows the calculated transport coefficient along the x and z directions with carrier concentration ranging from 3 to 3 electrons per unit

cell, the specific dopant types are not considered. Calculated rs of CrSi2 is plotted in Fig. 5(a). Obviously, the anisotropy of rs with ptype doping is larger than that with n-type doping. The anisotropy of rs with p-type doping increases with carrier concentration increasing, and rs along the z direction are much higher than that along the x direction. As shown in right panel of Fig. 2, for valence band, the largest dispersion is along the C–A (kz) direction (mz ¼ 0:14me ), and the smallest band dispersion is along the C– M (kx) direction (mx = 0.23 me). Therefore, for CrSi2, carrier along the z direction may have a higher mobility than those along the x direction. Consequently, rs along the z direction could be larger than that along the x direction. Thus, CrSi2 may have a large anisotropy of electrical conductivity with p-type doping. For the conduction bands of CrSi2, mz and mx are 0.42me and 0.20me, respectively. Hence, for n-type doping, the rs along the x direction should be larger than that along the z direction. Fig. 5(b) shows the anisotropy of Seebeck coefficient for CrSi2. As seen in this figure, the anisot-

(a)

250

S

200

200

(b)

0 2

σ/τ

(c)

1

(d) 2

S σ/τ

σ/τ

0 2

S σ/τ

10 5

5

0 6 5 4 3 2 1 0

(f)

15

10

n

n

15

2

(e)

S

250

400

800

1200

Temperture (K)

(g)

1 0 6 5 4 3 2 1 0

(h)

400

800

1200

Temperture (K)

Fig. 4. Calculated temperature-dependent transport properties of CrSi2 (left) and b-FeSi2 (right), respectively. (a) (e) Seebeck coefficients, S (unit in 106 V K1); (b) (f) Carrier concentration, n (unit in 1020 e/cm3); (c) (g) electrical conductivities relative to relaxation time, rs (unit in 1019 1/X ms); and (d) (e) powerfactor with respect to relaxation 2 time, S sr (unit in 1011 W/K2 ms).

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xx zz

(a) σ/τ 1

400

300K 800K 1000K 1500K

200

S

S

0.25

400

(b)

200 0

0

-200

-200

-400

-400

(c)

(e)

(f)

5

S σ/τ

15

2

10

2

S σ/τ

(d)

0.5

σ/τ

2

2.5

5 0

0 -3

-2

-1

0

1

2

3

-3

-2

-1

n (e/uc)

0

1

2

3

n (e/uc)

Fig. 5. Calculated transport coefficients of CrSi2 (left) and b-FeSi2 (right) as function of carrier concentration. (a) (d) Electrical conductivities relative to relaxation time, rs (unit 2 in 1020 1/X ms); (b) (e) Seebeck coefficients, S (unit in 106 V K1); and (c) (f) powerfactor with respect to relaxation time, S sr (unit in 1011 W/K2 ms).

ropy of Seebeck coefficient is not explicitly affected by the change of the temperature and carrier concentration, and the largest value 2 of S is 370 lV/K with light doping. The anisotropy of S sr for CrSi2 is shown in Fig. 5(c). From this figure, we can see that the anisotropy 2 of S sr for p-type doping may be larger than that of n-type doping. The possible reason is that the anisotropy of rs for p-type doping is larger than that of n-type doping. We also calculated the total density of states effective mass ðmDOS Þ to further check whether p-type doping is more beneficial for thermoelectric performance. The mDOS was calculated by mDOS ¼ ðm1 m2 m3 Þ1=3 N 2=3 v [34], where m1 ; m2 , and m3 are the effective mass components along three perpendicular direction, Nv is the band degeneracy. Our calculated mDOS of hole (0.26me) is smaller than that of electron (0.82me), which further proves that p-type doping may be better than n-type doping for achieving high thermoelectric performance of CrSi2. The 2 largest value of S sr along the z direction is 1.68  1012 W/K2 ms, and the high-temperature thermoelectric performance of CrSi2 can be improved by hole doping, 0.109–0.065 e/uc (1– 6  1021 cm3). Fig. 5(d) shows the anisotropy of rs for b-FeSi2. The anisotropy of r increases with the increasing of carrier concentration, and the s anisotropy of n-type could be larger than that of p-type doping. For p-type doping, rs along the x direction are slightly higher than those along the z direction. With increasing of the carrier concentration, rs along the x direction are slightly lower than those along the z direction. For n-type doping, rs along the z direction are always higher than those along the x direction. Moreover, for b-FeSi2, rs along the z direction with n-type doping is higher than that with 2 p-type doping, which may induce higher S sr with n-type doping than that with p-type doping, as shown in Fig. 5(f). As seen in the right panel of Fig. 2, for conduction bands, the band dispersion along C–Z (kz) direction mz ¼ 0:48 me is similar to the band dispersion along C–S (kx) direction mz ¼ 0:42 me . Compared with CrSi2, the anisotropy of electrical conductivity for b-FeSi2 is not so strong. Fig. 5(e) shows the anisotropy of Seebeck coefficient for b-FeSi2. As same as CrSi2, the anisotropy of Seebeck coefficient is not explicitly affected by the change of the temperature and carrier concentration, and the largest value of S is 450 lV/K with light doping. 2 Fig. 5(f) shows the anisotropy of S sr for b-FeSi2. We can see that 2 the anisotropy of S sr for n-type doping is probably larger than that 2 of p-type doping, and the largest value of S sr along the z direction is

2

0.59  1012 W/K2ms. Both CrSi2 and b-FeSi2, the anisotropy of S sr could be due to the anisotropy of rs. 4. Conclusion Using first-principles calculations and semiclassical Boltzmann theory, we investigated the transport properties and electronic structures of CrSi2 and b-FeSi2. We find that the anisotropy of the transport properties mainly comes from the anisotropy of the band structure. The calculated band structures show that CrSi2 and bFeSi2 are both narrow gap semiconductors. Both CrSi2 and b-FeSi2 may have high Seebeck coefficient (above 170 lV/K) from 300– 1500 K. Moreover, their carrier concentration increases rapidly with the increasing of temperature, which may lead to a high electrical conductivity in high temperature range. High electrical conductivity and considerable Seebeck coefficient may induce high thermoelectric performance of CrSi2 and b-FeSi2 in high temperature range. For CrSi2, p-type doping may achieve higher thermoelectric performance than n-type doping, which mainly results from the high anisotropy of its electrical conductivity with p-type doping. However, for b-FeSi2, n-type doping could be more beneficial for thermoelectric performance than p-type doping, which probably comes from its relative large electrical conductivity with n-type doping. Acknowledgments This research was sponsored by the National Natural Science Fundation of China (No. 21071045), the Program for New Century Excellent Talents in University (No. NCET-10-0132), and Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No. 13IRTSTHN017). References [1] G. Mahan, B. Sales, J. Sharp, Phys. Today 50 (1997) 42. [2] G. Slack, in: D.M. Rowe (Ed.), CRC Handbook of Thermoelectrics, CRC Press, Boca Raton, 1995, pp. 407–440. [3] Y.L. Yan, Y.X. Wang, J Mater. Chem. 21 (2011) 12497. [4] Y.L. Yan, Y.X. Wang, G.B. Zhang, J Mater. Chem. 22 (2012) 20284. [5] A.K. Vasudevan, J.J. Petrovic, Mater. Sci. Eng. A 155 (1992) 1.

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